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## Algebra 1

### Course: Algebra 1 > Unit 6

Lesson 3: Solving systems of equations with elimination- Systems of equations with elimination: King's cupcakes
- Elimination strategies
- Combining equations
- Elimination strategies
- Systems of equations with elimination: x-4y=-18 & -x+3y=11
- Systems of equations with elimination
- Systems of equations with elimination: potato chips
- Systems of equations with elimination (and manipulation)
- Systems of equations with elimination challenge
- Elimination method review (systems of linear equations)

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# Systems of equations with elimination: King's cupcakes

Sal uses simple elimination to figure out how many cupcakes are eaten by children and adults. Created by Sal Khan.

## Want to join the conversation?

- I understand the math and the method. I graphed all of the problems with no trouble and I got the correct answers from the x and y axes. What I don't understand is how this works. It seems almost magical. Two lines intersect and Ka-Boom, you have the two solutions. I feel I am missing something important but I can't see it. That's why It seems magical.

Thanks for your insight!!(53 votes)- When you solve a system of equations, the whole purpose is to find how the 2 equations relate to each other, and whether or not they have a common solution. The common solution is the point where the 2 lines intersect because it is a point that sits on both lines.(42 votes)

- I love to drink my cupcakes (7:10)(48 votes)
- I don't understand why Sal over complicated this. Within the first 5 seconds of seeing " 500a + 200c = 2900" and "500a + 300c = 3100", I knew the answer. I found this by observing the difference between the two equations. Let me explain, in both equations, 500a does not change, it stays the same throughout. which means that the only difference between the two equations is that the amount of C's or children.

Because of that, I was able to tell that 1c =2, because adding 100c, with A (or Adults) remaining static, added 200 to the final product. which mean that c was a variable of 2. Once I knew this, I went back to the first equation(500a + 200c = 2900), plugged in 2 for c, subtracted 400 from 2900, then divided 2500 by 500, which equals 5, the variable of A. So I ask you, is this a special occurrence that just happens to work for a few equations? Or if not, why not just use this uncomplicated way, which you can do in your head?(6 votes)- Because he is teaching an algebraic method to solve a problem. Different people use different methods to solve equations. This is part of an 8th grade math curriculum enhancement. So while it seems easier to solve in your head, it is only that easy once you have learned how to do it....like he explained here :)(33 votes)

- Just to let you know Sal can draw like crazy. Does he only use the computer? I tried but my work is all sloppy!(16 votes)
- You can use specialized drawing tablets with a stylus to imitate drawing on a real paper, which is how all of these videos are made.(11 votes)

- Ok, reading through the comments it seems like there a lot of diffewrent ways to solve systems of equation. Can someone give me a quick list? I would like to try other ways to see if I can wrap my head around it easier.

P.S.

I don't mind if there are no videos for it on Khan Academy as long as I am reffered to a explanation.(9 votes)- I only know of four, substitution, elimination, graphing, and using a matrix (this does not count letting a calculator do all the work for you). Sal shows all 4 I think, but matrices are not in the Algebra I section.(10 votes)

- Sal is definitely human, and not a robot, cause humans make mistakes.
**children drink two cups**...(7 votes)- that was epic lol i was so focused and serious in one second even my headache is gone lol(5 votes)

- why is adult and child add up together 2900??(4 votes)
- Because each adult ate 5 cupcakes,and each child ate 2. There were 500 adults and 200 children. So if each adult ate 5 cupcakes and there were 500 adults,then you would multiply 500 by five. And it's the same with the children. If each child ate 2 cupcakes and there were 200 children,then you would multiply 200 by 2. And then you add the two answers together,and you have 2900.(5 votes)

- is there any other ways to solve this specific equation that is harder or easier in factor?(4 votes)
- I watched this video attentively, but solved the problem three ways before seeing how Sal did it here. I graphed it but only with considerable difficulty, as I just didn't get it to begin with and had to slowly figure it out, fortunately solidifying what I knew about linear equations along the way. I was initially confused by the lack of immediately apparent relation between the total numbers of cupcakes and the y-intercept, and by the fact that both slopes were negative whereas I was thinking of ascending lines. But I finally understood it better and wound up with a very nice graph with the intersecting point exactly at (5,2).

Then I had further problems with substitution, also owing simply to initial confusion but also – worse – because I made a mistake, writing 10 1/2 rather than 10 1/3 when I substituted. It took me too long to find out where the problem was, but I finally caught it. My original simplified equations were all right and the substitution worked.

Before I did either of these, however, I quickly solved the problem very easily without using anything I knew as a method at all. When there were 100 more children, there were 200 more cupcakes. Therefore, obviously, each child ate 2 cupcakes and c = 2. When I look for a way to explain what I did, it looks like though I didn't give conscious attention to the 500a's, what I in fact did was to subtract the smaller party from the larger one: 500a - 500a = 0 (though I didn't even think about that), 300 - 200 = 100, 3100 - 2900 = 200, and then the solution is obvious.

Now, it looks to me like Sal did essentially the same thing, except he appears to have subtracted the larger party from the smaller one (6:30). Except he's not talking about subtracting anything from anything generally, but rather about a process of "elimination" that I may need another video or two to fully grasp.

My question is, what about my "subtraction" method? It seems to be a bit different from the other three (?), but it yields the same answer and I'm wondering how it could be characterized. It might imaginably be the same as substitution or elimination, but it seems simpler. I realize it may only work under certain limited conditions, specifically when one of the values is equal such as 500a is here. But does it constitute a special case, if only in this circumstance? Thanks.(3 votes)- Khan Academy has a unit with a few videos focusing on it. The unit can be found here: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations#x2f8bb11595b61c86:equivalent-systems-equations

I'd recommend the "Why we can subtract One Equation from the Other in a Systems of Equations". However, if you have time, I'd recommend watching all the videos and doing the tests, as they are very informative, and give evidence/proof to why some methods work.

It seems like your spending a lot more time than you should be on these problems, because your making a small mistake here and there, which ends up costing you time. This isn't a big deal, but when you get into higher classes, you want eliminate these minor mistakes. So I'd recommend reading the problem fully, and looking at each step after you do it, to see if it makes sense. So take your time, look at every step carefully, and try to eliminate these minor mistakes.

If you want more helps on Elimination: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations#x2f8bb11595b61c86:solving-systems-elimination

You didn't say much about substitution but just in case if you need help: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations#x2f8bb11595b61c86:solving-systems-of-equations-with-substitution

I hope that helps.(5 votes)

- can you subtract the orange from blue instead ? Why doesn't sal do that(4 votes)
- Yes, and if you chose that way, it may be easier because you would not have to deal with negative numbers by doing it that way as I suspect you might be thinking.(3 votes)

## Video transcript

After you cross the troll's
bridge and you save the prince or princess, you return them
back to their father, the King. And he's so excited that you
returned their son or daughter to him that he wants to
throw a brunch in your honor. But he has a little
bit of a conundrum in throwing the brunch. He wants to figure out how
many cupcakes should he order? He doesn't want to
waste any, but he wants to make sure that
everyone has enough to eat. And you say, well
what's the problem here? And he says, well I know
adults eat a different number of cupcakes than children eat. And I know that in my
kingdom, an adult will always eat the same amount
and a child will always eat the same amount. And so you say, King, well what
information can you give me? I might be able to help
you out a little bit. You're feeling very confident
after this troll situation. And he says, well I know at the
last party we had 500 adults and we had 200
children, and combined they ate 2,900 cupcakes. And you say, OK,
that's interesting. But I think I'll need a
little bit more information. Have you thrown
parties before then? And the king says,
of course I have. I like to throw parties. Well what happened at
the party before that? And he says, well there
we also had 500 adults and we had 300 children. And you say, well
how many cupcakes were eaten at that party? And he says, well we know
it was 3,100 cupcakes. And so you get a
tingling feeling that a little bit of algebra
might apply over here. And you say, well let me see. What do we need to figure out? We need to figure out the
number of cupcakes on average that an adult will eat. So number of cupcakes
for an adult. And we also need to
figure out the number of cupcakes for a child. So these are the
two things that we need to figure out
because then we can know how many adults
and children are coming to the next branch that are
being held in your honor and get the exact right
number of cupcakes. So those are things you're
trying to figure out. And we don't know
what those things are. Let's define some variables
that represent those things. Well let's do a for adults. Let's let a equal the
number of cupcakes that, on average,
each adult eats. And let's do c for children. So c is the number of cupcakes
for a child on average. So given that
information, let's see how we can represent what the
King has told us algebraically. So let's think about this
orange information first. How could we represent
this algebraically? Well let's think about
how many cupcakes the adults ate at that party. You had 500 adults, and
on average, each of them exactly a cupcakes. So the total number of
cupcakes that the adults ate were 500 times a. How many did the children eat? Well same logic. You had 200 children and
they each ate c cupcakes. So 200 times c is the
total number of cupcakes that the children ate. Well how much did
they eat in total? Well it's the total
number that the adults ate plus the total number
that the children ate which is 2,900 cupcakes. So let's do that and apply that
same logic to the blue party right over here, this
blue information. How can we represent
this algebraically? Well once again, how many total
cupcakes did the adults eat? Well, you had 500
adults and they each ate a cupcakes, which
is an unknown right now. And then what
about the children? Well you had 300 children
and they each ate c cupcakes. And so if you add
up all the cupcakes that the adults ate plus all the
cupcakes that the children ate, you get to 3,100 cupcakes. So this is starting
to look interesting. I have two equations. I have a system of two
equations with two unknowns. And you know from your
experience with the troll that you should be
able to solve this. You could solve it graphically
like you did in the past, but now you feel
that there could be another tool in your
tool kit which is really just an application of the
algebra that you already know. So think a little bit about
how you might do this. So let's rewrite this first
equation right over here. So we have 500a plus 200c
is equal to 2,900 cupcakes. Now, it would be good if we
could get rid of this 500a somehow. Well you might say, well
let me just subtract 500a. So you might say, oh, I
just want to subtract 500a. But if you subtracted 500a
from the left hand side, you'd also have to subtract
500a from the right hand side. And so the a wouldn't
just disappear. It would just end up
on the right hand side, and you would still
have one equation with two unknowns which
isn't too helpful. But you see something
interesting. You're like, well
this is a 500a here. What if I subtracted
a 500a and this 300c? So if I subtracted
the 500a and the 300c from the left hand side. And you're like, well
why is that useful? You're going to do the same
thing on the right hand side and then you're going to have
an a and a c on the right hand. And you just say, hold on,
hold on one second here. Hold on, I guess you're
talking to yourself. Hold on one second. I'm subtracting the left
hand side of this equation, but this left hand side
is the exact same thing as this right hand side. So here I could
subtract 500a and 300c, and I could do 500a and
subtract 300c over here. But we know that
subtracting 500a and 300c, that's the exact same
thing as subtracting 3,100. Let me make it clear. This is 500a minus 300c
is the exact same thing as subtracting 500a plus 300c. And we know that 500a plus
300c is exactly 3,100. This is 3,100. This is what the second
information gave us. So instead of subtracting
500a and minus 300c, we can just subtract 3,100. So let me do that. This is exciting. So let me clear that out. So let's clear that out. And so here, instead
of doing this, I can subtract the exact same
value, which we know is 3,100, subtract 3,100. So looking at it
this way, it looks like we're subtracting
this bottom equation from the top equation, but
we're really just subtracting the same thing from both sides. This is just very
basic algebra here. But if we do that,
let's see what happens. So on the left hand side, 500a
minus 500a, those cancel out. 200c minus 300c, that
gives us negative 100c. And on the right hand
side, 2,900 minus 3,100 is negative 200. Well now we have one
equation with one unknown, and we know how to solve this. We can divide both
sides by negative 100. These cancel out. And then over here, you
end up with a positive 2. So c is equal to positive 2. So we've solved one
of the unknowns, the each child on
average drinks two cups. So c is equal to 2. So how can we figure
out what a is? Well now we can take
this information and go back into either one
of these and figure out what a has to be. So let's go back into the
orange one right over here and figure out what a has to be. So we had 500a plus 200c, but
we know what c is, c is 2. So 200 times 2 is
equal to 2,900. And now we just
have to solve for a, one equation with one unknown. So we have 500a,
200 times 2 is 400, plus 400 is equal to 2,900. We can subtract 400 from
both sides of this equation. Let me do that. Subtracting 400, and we are
left with this cancels out. And on the left hand
side, we have 500a. This is very exciting. We're in the home stretch. On the right hand
side, you have 2,500. 500a equals 2,500. We can divide both
sides by 500, and we are left with 2,500
divided by 500 is just 5. So you have a is equal
to 5 and you're done. You have solved the
King's conundrum. Each child on average drinks
2 cups of water-- sorry, not cups of water. I don't know where
I got that from. Each child 2 cupcakes and each
adult will eat 5 cupcakes. a is equal to 5. And so based on how many
adults and children are coming to the brunch in
your honor, you now know exactly the
number of cupcakes that the King needs to order.